Analyses of Non-Stationary Schemes

UP is definitely an exception. Most of the interesting non-stationary schemes can be looked at as letting the coefficients of a scheme take some trajectory in the design space of fixed finite dimension (and fixed arity) considered above. Except when the coefficient of the widest box-spline happens to drop to zero, the support will remain constant, at that given by the widest scheme included in the linear combinations. 

The design of a non-stationary scheme can be regarded then as the design of a trajectory in design space. 

Such a trajectory has an important property. It could converge towards a limiting scheme, it could repeat itself, or it could wander around randomly. Discount the last possibility. If it repeats itself after a fixed number of steps, then we can take that cycle of steps together as a single scheme of high arity and discover that it is essentially a stationary scheme. 

If it does converge to a limit scheme, then that convergence is an important property for those criteria which depend most on the late stages. 

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